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Mathematical Medicine and Biology 1998 15(4):367-385; doi:10.1093/imammb/15.4.367
© 1998 by Institute of Mathematics and its Applications
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Stability analysis of the FitzHugh–Nagumo differential equations driven by impulses: Applied to the electrical firing of magnocellular neurons

ALICE E. MILNE{dagger} and ZAID S. CHALABI

Silsoe Research Institute Wrest Park, Silsoe, Bedford MK45 4HS, UK

{dagger} Email: alice.milne{at}bbsrc.ac.uk and zoid.chalabi{at}bbsrc.ac.uk

A stability analysis is carried out for a mathematical model which describes the electrical firing of a single vasopressin neuron. The model used is a FitzHugh–Nagumo-type system which is driven by impulses. The analysis is based on recent developments in the stability theory of impulsive differential equations. Conditions are derived under which the system of differential equations is stable at two of its equilibrium points. Biologically this bistability represents the cell alternating between periods of electrical activity and silence. The conditions for stability are specified in terms of the amplitude and frequency of the impulses perturbing the system. Both stochastic and deterministic impulses are considered.

Keywords: FitzHugh–Nagumo equation; nerve condition; impulsive differential equations; stability analysis; integral inequalities


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